The present invention relates to a method for encoding computer-generated holograms in pixelated light modulators whose encoding surface comprises a pixel matrix whose pixels have a certain pixel shape and pixel transparency, where the encoding surface comprises a hologram which is composed of sub-holograms each of which representing one object point of the object which is to be reconstructed by the hologram, where a pyramidal body with a virtual observer window as the defined visibility region and with the object point as the peak is extended beyond the object point and projected onto the encoding surface, thus creating an encoding region in which the object point is holographically encoded as a sub-hologram.
Light modulators with their encoding surfaces are either of a transmissive or reflective type, and they comprise a matrix of pixels with finite extent, which are separated by more or less wide gaps owing to the manufacturing process. In the case of a liquid crystal modulator, the encoding surface is e.g. crossed by a grid of thin electrodes, where the grid represents a matrix of electrodes which intersect at right angles, thus defining rectangular regions between the electrodes, the so-called pixels, which are disposed at a certain distance to each other, the so-called pixel pitch p. The matrix of electrodes is also known as inter-pixel matrix or gap grid, because it exhibits gaps g between the pixels. It can be switched with the help of an electronic controller, in particular with the help of a computer by software means, in order to encode the pixels as regards their amplitude and/or phase such that they exhibit a certain transmittance or reflectance. Pixels which are encoded as transmissive pixels let the incident waves pass, while the pixels which are encoded as reflective pixels reflect the incident waves.
A method for calculating computer-generated video holograms and a corresponding device are known from document DE 10 2004 063 838 A1, where object points with complex amplitude values of a three-dimensional original object are assigned to matrix dots of parallel virtual object section planes in order to define for each object section plane a separate object data set with discrete amplitude values in the form of matrix dots of a given matrix, and to calculate from the object data sets a holographic code for the pixel matrix of a light modulator.
For this, a diffraction pattern is computed in the form of a separate two-dimensional distribution of wave fields for a reference plane, which is situated at a finite distance and parallel to the object section planes, from each object data set of each object section plane, where the wave fields of all object section planes are computed for at least one common virtual observer window which is situated in the reference plane near the eyes of an observer, and whose window area is reduced compared with the hologram.
The calculated distributions for the wave fields of all object section planes are added in a reference data set in order to define an aggregated wave field for the virtual observer window. For generating a hologram data set for the common computer-generated hologram of the object, the reference data set is transformed into a hologram plane, which is situated at a finite distance and parallel to the reference plane that coincides with the plane of the pixel matrix of the light modulator.
The amplitude and phase values of the hologram, which are to be realised in the individual pixels, are also calculated dot by dot for the hologram plane. Typically, two-dimensional light modulators with an encoding surface of m pixel rows at n pixels each are used for recording computer-generated holograms, where the pixels are no points, but have a finite extent and a given shape and a certain amplitude transparency and phase transparency.
One problem of the prior art is that the point-wise computation of the hologram and its representation in pixels with finite extent on the light modulators cause the hologram to be biased and the corresponding instances of imprecision in the visible reconstruction to be perceived by the observer.
The occurring defects are caused by the real extent of the pixels and are based on a conflict between the point-wise computation of the hologram and the real extent of the pixels, which remains unconsidered.
It is also known that the e.g. rectangular pixels of the light modulator, given a uniform transmittance or reflectance, exhibit an amplitude distribution in the form of a sinc function of
      sin    ⁢                  ⁢          c      ⁡              (        x        )              =            sin      ⁡              (                  π          ⁢                                          ⁢          x                )                    π      ⁢                          ⁢      x      in a Fourier plane when they are illuminated with coherent light.
The computation of the complex light distributions in the plane of the observer window and in the hologram plane only applies to points which are intersecting points of a given virtual grid. If the complex distributions are represented on a light modulator, then there are pixels which have for example a rectangular shape and which exhibit a constant amplitude and/or phase transparency, as said above. The representation of the complex hologram values in the pixels of a real light modulator is mathematically a convolution of the computed hologram with a rectangular function that represents the pixel extent in the x and y direction. This mathematical process known as convolution causes—during the reconstruction of the hologram—the Fourier transform of the ideal hologram, which is encoded point-wise, to be multiplied with a sinc function which is the Fourier transform of the pixel function, which is a rectangle, in the plane of the observer window. An observer who watches the reconstruction of the object thus perceives this defect.
Serving as a visibility region for an observer in the reference plane, i.e. in a virtual plane which lies immediately in front of the observer eye, an observer window has a given size; it can for example be as large as an eye pupil or be somewhat larger than that, e.g. have twice or three times the size of an eye pupil.
One problem is that the complex wave front in the given observer window and thus also the reconstruction of the three-dimensional object in the space between the observer window and the hologram are biased by the effects of the finite pixel extent in the light modulator, in that for example undesired changes in intensity may occur in the observer window. If the observer window is larger than the eye pupil, then for example the reconstruction of the three-dimensional object appears darker to an observer whose eye pupil is situated near the edge of the observer window than to an observer whose eye pupil is situated in the centre of the observer window. In addition to changes in brightness, there is also noise, i.e. a deterioration in quality of the reconstruction of the three-dimensional scene.
In U.S. Application No. 12/440,478, a hologram computation method based on document DE 10 2004 063 838 A1 is described where a correction is carried out with an inverse or reciprocal of the transform of the pixel shape and the pixel transparency in the observer plane. This requires knowledge of the complex values of the wave front in the observer plane. Fourier transforms are required for this computation.
Document WO2004/044659 A2 describes a device for reconstructing video holograms in which a holographic encoding takes place, as shown in FIG. 1. The three-dimensional object 10 is composed of object points, of which two object points 30, 31 are shown in the drawing. Pyramidal bodies with the observer window 11 as the base and the two selected object points 30, 31 of the object 10 as the respective peaks are extended beyond these object points 30, 31 and projected onto the encoding surface with the desired final hologram 12. Thereby, object-point-related encoding regions 20, 21 in which the object points 30, 31 can be holographically encoded in respective sub-holograms 201, 211 are created on the given encoding surface.
The total hologram is then the complex-valued sum of all sub-holograms. Mainly those sub-regions of the encoding surface which correspond with these encoding regions 20, 21 contribute to the reconstruction of individual object points 30, 31 of the three-dimensional object. The computer-generated holograms 12 are illuminated with an illumination system with an array of micro-lenses 15 to generate the reconstruction.
Holograms for such a device for reconstructing video holograms can be computed according to the method described in document DE 10 2004 063 838 A1.
Another method for computing holograms is described in U.S. Pat. No. 8,437,056, which discloses an analytical computation of sub-holograms on the encoding surface of a light modulator in the form of lens functions. Thereafter, the sub-holograms are added to form a total hologram.
Within the section of the total hologram which is defined by the encoding regions 20, 21, the individual sub-holograms have a substantially constant amplitude whose value is determined depending on brightness and distance of the object points and a phase which corresponds with a lens function, where the focal length of the lens and the size of the encoding regions are variable depending on the depth coordinate of the object point. Outside the section which is defined by the encoding regions 20, 21, the amplitude of the respective sub-holograms is 0. The total hologram is then the complex-valued sum of all sub-holograms. In the case of point-shaped pixels, the virtual observer window would be created based on the total hologram by a Fourier transform or, optionally, by a different transformation, such as a Fresnel transform.
However, for the computation of the hologram according to this method, the wave front in the observer window is not found explicitly mathematically. The method does not use any Fourier or Fresnel transform. The computation thus has the advantage that it takes less computing time compared with the method described in document DE 10 2004 063 838 A1.
The problem is that also in this method for computing holograms the pixel shape and the pixel transparency of the light modulator are not taken into consideration.